The champions moved up to second spot after nine matches pending the weekend games.
The Ghana midfielder has barely featured under new coach Rafael Benitez this term but after coming on for the injured Esteban Cambiasso, he unleashed a 25-metre drive on the stroke of halftime which goalkeeper Eduardo somehow let in.
The Portugal keeper redeemed himself slightly in the second half when denying Jonathan Biabiany's volley while Genoa's Luca Toni was a constant menace as the hosts missed several chances.
"It's a very important victory," Inter's injury-prone left-back Davide Santon told Sky having filled in for the stricken Cristian Chivu.
"I'm also very happy for myself, finally I feel good and I've managed to finish a game. We suffered because of their intensity, it was tough."
Muntari stormed out of the San Siro after not even being named on the bench for the 0-0 draw with Juve earlier this month and his Inter future looked bleak.
His last goal for the club was October 24 last year.
Inter's joy in one of Serie A's most intimidating stadiums, where Italy's Euro 2012 qualifier was abandoned earlier this month when Serbia fans rioted, was mitigated slightly by the early injury to Cambiasso and keeper Julio Cesar also going off.
The European champions travel to Tottenham Hotspur in the Champions League on Tuesday, the reason for Serie A's introduction of a Friday match, but they go to England with a busy treatment room.
Forwards Diego Milito and Goran Pandev were only fit enough for the bench in Genoa but will be pushing for starts at Spurs despite the decent form of young stand-ins Biabiany and Brazil's Coutinho.
Inter's resilience will have pleased Benitez given they were almost embarrassed by Tottenham last week when they won 4-3 having been 4-0 up, while they drew 1-1 at home to Sampdoria at the weekend after a patchy display.
